Representations for the extreme zeros of orthogonal polynomials

We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth death process (with killing) are displayed

Representations for the extreme zeros of orthogonal polynomials

We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth death process (with killing) are displayed

On Probability Characteristics for a Class of Queueing Models with Impatient Customers

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Bounds and asymptotics for the rate of convergence of birth-death processes

We survey a method initiated by one of us in the 1990's for finding bounds and representations for the rate of convergence of a birth-death process. We also present new results obtained by this method for some specific birth-death processes related to mean-field models and to the $M/M/N/N+R$ service system. The new findings pertain to the asymptotic behaviour of the rate of convergence as the number of states tends to infinity

On the speed of convergence to stationarity of the Erlang loss system

We consider the Erlang loss system, characterized by $N$ servers, Poisson arrivals and exponential service times, and allow the arrival rate to be a function of $N.$ We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and display some bounds for the total variation distance between the time-dependent and stationary distributions. We also pay attention to time-dependent rates

Tot allò que sempre has volgut saber sobre la Viquipèdia

We correct representations for the endpoints of the true interval of orthogonality of a sequence of orthogonal polynomials that were stated by us in the Journal of Computational and Applied Mathematics 233 (2009) 847–851

The decay function of nonhomogeneous birth-death processes, with application to mean-field models

The paper develops in different directions the method of the second author for estimation of the rate of exponential convergence of nonhomogeneous birth-death processes. Applying the method to mean-field models, we discover some phenomena related to their spectral gaps.Birth-death processes Spectral gap Mean-field models Critical points Decay function

Birth-death processes with killing

The purpose of this note is to point out that Karlin and McGregor's integral representation for the transition probabilities of a birth-death process on a semi-infinite lattice with an absorbing bottom state remains valid if one allows the possibility of absorption into the bottom state from any other state. Conditions for uniqueness of the minimal transition function are also given

Extinction probability in a birth-death process with killing

We study birth-death processes on the nonnegative integers, where {1, 2,...} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator. \u